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Locality and Causality in Classical Field Theory

Locality and Causality have long been an essential aspect of axiomatizations of Quantum Field Theories (Wightman, Haag-Kastler, Brunetti-Fredenhagen-Verch). Unfortunately, these axioms explicitly rely on a fixed background Lorentzian metric, thus excluding theories with dynamical causal structure such as General Relativity. The same problem persists even when the axioms are adapted to Classical Field Theory. On the other hand, dynamical causal structure has long been understood in the classic literature on quasilinear hyperbolic PDEs. I will describe how the translation of well-posedness properties of quasilinear hyperbolic PDEs into an appropriate algebraic language essentially generalizes existing Locality and Causality axioms and turns them into theorems of Classical Field Theory, while also suggesting a generalization to Quantum Field Theory. [based on arXiv:1211.1914]